The proof of Theorem REF  can be found in Appendix A.2. Theorem REF  provides nearly sufficient and necessary conditions for numerical convergence.
Specifically, from Theorem REF (a), we know that to ensure the numerical convergence, we must have \(0 \le \gamma <1\) .
This corroborates the common practice in the existing literature very well [1]}.
Otherwise, by Theorem REF (b), we know that the numerical convergence can never be achieved if \( \gamma >1 \) .
Consequently, the FMGDM estimator does not exist in this case.
Furthermore, with a given momentum parameter \(0\le \gamma <1\) , the learning rate \( \alpha \)  should be sufficiently small.
Otherwise, the algorithm should not converge.
Remarkably, the upper bound of the learning rate \( \alpha \)  is related to the largest eigenvalue of \( \Sigma _{xx} \)  (i.e., \( \lambda _{1} \) ) and the momentum parameter \( \gamma \) .
This upper bound should be larger if the momentum parameter \( \gamma \)  is larger.
As a result, this enables a larger learning rate (thus faster convergence) to be used for a FMGDM algorithm compared to a standard GD algorithm (\( \gamma =0 \) ).
Theorem REF  examines the conditions for the numerical convergence for the FMGDM algorithm.
However, it does not provide the exact convergence rate.
In particular, we want to understand how the \( \alpha \)  and \( \gamma \)  specification would affect the numerical convergence rate.
We next develop the following theorem.
